As presented so far, the full history I-state is needed to determine a derived I-state. It may be preferable, however, to discard histories and work entirely in the derived I-space. Without storing the histories on the machine or robot, a derived information transition equation needs to be developed. The important requirement in this case is as follows:
If is replaced by , then must be correctly determined using only , , and .
Whether this requirement can be met depends on the particular I-map. Another way to express the requirement is that if is given, then the full history does not contain any information that could further constrain . The information provided by is sufficient for determining the next derived I-states. This is similar to the concept of a sufficient statistic, which arises in decision theory . If the requirement is met, then is called a sufficient I-map. One peculiarity is that the sufficiency is relative to , as opposed to being absolute in some sense. For example, any I-map that maps onto is sufficient because is always known (it remains fixed at 0). Thus, the requirement for sufficiency depends strongly on the particular derived I-space.
For a sufficient I-map, a derived information transition equation is determined as
The diagram in Figure 11.4a indicates the problem of obtaining a sufficient I-map. The top of the diagram shows the history I-state transitions before the I-map was introduced. The bottom of the diagram shows the attempted derived information transition equation, . The requirement is that the derived I-state obtained in the lower right must be the same regardless of which path is followed from the upper left. Either can be applied to , followed by , or can be applied to , followed by some . The problem with the existence of is that is usually not invertible. The preimage of some derived I-state yields a set of histories in . Applying to all of these yields a set of possible next-stage history I-states. Applying to these may yield a set of derived I-states because of the ambiguity introduced by . This chain of mappings is shown in Figure 11.4b. If a singleton is obtained under all circumstances, then this yields the required values of . Otherwise, new uncertainty arises about the current derived I-state. This could be handled by defining an information space over the information space, but this nastiness will be avoided here.
Since I-maps can be defined from any derived I-space to another, the concepts presented in this section do not necessarily require as the starting point. For example, an I-map, , may be called sufficient with respect to rather than with respect to .
Steven M LaValle 2020-08-14